document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210202>Question 1210202</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851595 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851595\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The AI solution from the other tutor is wrong; a subset of the integers from 0 to 9 inclusive can't contain the same element twice.<br><BR>\n" );
document.write( "The requirements are that the subset must contain at least two elements and the sum of the two largest elements is 4.<br><BR>\n" );
document.write( "(1) If the largest element in the subset is 4 and the sum of the two largest elements is 4, then the second largest element in the subset is 0.  Then, since 0 and 4 are the two largest elements, there can be no other elements in the subset.<br><BR>\n" );
document.write( "So the subset {4,0} satisfies the conditions.<br><BR>\n" );
document.write( "(2) If the largest element in the subset is 3 and the sum of the two largest elements is 4, then the second largest element in the subset is 1.  Then, since 3 and 1 are the two largest elements, the subset can either contain or not contain the element 0.<br><BR>\n" );
document.write( "So the subsets {3,1} and {3,1,0} satisfy the conditions.<br><BR>\n" );
document.write( "There are no other subsets that satisfy the conditions.<br><BR>\n" );
document.write( "ANSWER: There are three subsets that satisfy the conditions:<BR>\n" );
document.write( "{4,0}<BR>\n" );
document.write( "{3,1}<BR>\n" );
document.write( "{3,1,0}<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );